european synchrotron radiation facility · why neutrons? page 2 9 july 2014 l christian vettier...

WHAT DO WE STUDY WITH NEUTRONS?

9 July 2014 l Christian Vettier ESRF-Grenoble & ESS-Lund

2010

Materials for energy, heath, environment archaeological artefacts, commercial products

phase transitions

magnetic orderings

magnetic fluctuations

exchanges bias

soft matter

polymers

2000

1990

1980

1970

1960

WHY NEUTRONS?


Why do we use neutrons?

- Neutrons tell us about the positions and motions of atoms/magnetic moments in condensed matter

- Neutrons interact with nuclei and magnetic moments

the two interactions have similar ‘strengths’

-Interaction with matter is gentle and simple:

scattering data are easy to interpret

- Neutrons are penetrating: bulk materials can be studied

any sample can be contained in special environment

- Experimental science: instrument design, data taking and data analysis ……

WHAT DO WE MEASURE?


Scattering experiments: neutrons in and neutrons out!

We measure the number of neutrons scattered by a sample

against the number of incident neutrons (neutron flux)

• Scattered intensities involve positions and motions of scattering centres: atoms/magnetic moments • Scattered intensities are proportional to Fourier transforms (in space and time) pair correlation functions

as a function of the change in direction and energy of the scattered neutrons

as a function of polarisation or polarising magnetic fields

source of neutron beams

neutron spectroscopy

Inelastic scattering

HOW DO WE MEASURE?


source of neutron beams

neutron diffraction

OVERVIEW


• a bit history

• neutron properties

• interactions between neutrons and matter

• measured quantities

• scattering by atoms/nuclei

• scattering by magnetic moments

• key messages

NEUTRONS: INTRODUCTION


A bit of history:

W. Bothe & H. Decker -1930

discovered very penetrating radiation emitted when α particles hit light elements

I. Curie & F. Juliot -1932

observed creation of p+ in paraffin sheets & thought new radiation was γ-rays

J. Chadwick -1932 a few months later

discovers the ‘neutron’, a neutral but massive particle

Nobel Prize in Physics

24He + 4

9Be → 616C + 0

1n

mHe + mB( )c2 + THe = mN + mn( )c2 + TN + Tn

24He + 5

10B → 714N + 0

1n

mn =1.0067 ± 0.0012 a.m.u



A bit of history:

E. Fermi showed that neutrons moderated by paraffin could be captured by various elements, producing artificial radioactive nuclei importance of neutron energy range

D.P. Mitchell & N. Powers / H. v. Halban & P. Preiswerk -1936

showed that thermal neutrons can be diffracted by crystalline matter

MgO crystals oriented (200) planes 22˚corresponds to Bragg angle for peak

of wavelength distribution of thermal neutrons ~0.16nm



A bit of history:

• O. Hahn, F. Strassmann & L. Meitner -1938

discovered the fission of 235U nuclei through thermal neutron capture

• H. v. Halban, F. Joliot & L. Kowarski -1939

showed that 235U nuclei fission produced 2.4 n0 on average – chain reaction • E. Fermi & al. -1942

first self-sustained chain reaction reactor CP1 Chicago

• C.G. Shull & B.N. Brockhouse -1994

Nobel Prize in Physics

• C.G. Shull -1942

Proof of antiferromagnetic order in MnO

NEUTRONS: NEUTRON PROPERTIES


free neutrons are unstable: β-decay proton, electron, anti-neutrino

life time: 886 ± 1 sec

wave-particle duality: neutrons have particle-like and wave-like properties

• mass: mn = 1.675 x 10-27 kg = 1.00866 u. (unified atomic mass unit)

• charge = 0

• spin =1/2 magnetic dipole moment: μn = -1.913 μN

• velocity (v) kinetic energy (E) temperature (T) wavevector (k) wavelength (λ)

E = mn v2 /2 = kBT = hk /2π( )2 /2mn

k = 2π / λ = mn v / h /2π( )

λ (nm) = 395.6/ v m/s( ) = 0.286/ E( )1/2 E in eV( )

E meV( ) = 0.02072 k2 (k in nm-1)

1(A0) ≈ 82 meV ≈ 124THz ≈ 950 K

T =Lv

= 252.77 µsec ⋅ λ Ao

⋅ L m[ ]

fortunately large value! monochromatisation: diffraction or time of flight

Example λ = 4 Å v=1000m/s E= 5 meV



Conversion chart

P. A. Egelstaff ed. - Thermal Neutron Scattering Academic Press 1965



Neutron energy ranges

ULTRA-COLD NEUTRONS


v ≈ 20m/s Ekin ≈ 2 µeV T= 0.023 K

λ = 200 Å

effect of gravity - neutrons are massive!

mirror ∼ potential well for ultra-cold neutrons

neutrons are ‘stacked’ at distinct height levels (in the micrometer range!)

note: cold neutron beams are bent by gravity ~ 1.2 cm at 100 m for 20 Å neutrons

the very cold side

neutrons : objects to study fundamental interactions

neutron β-decay

free neutrons are not forever

KEY MESSSAGES


• neutrons are not elementary particles

• they are not for ever

• neutrons are not only powerful probe, they can be studied as objects

NEUTRON SCATTERING: INTERACTIONS


Neutrons

X-rays

Electrons

Neutron scattering exploits ‘cool’ neutrons: 0.05 meV< En < 500 meV

0.4 Å < λ < 40 Å

NEUTRON SCATTERING: INTERACTIONS


Neutrons interact with nuclei:

capture: absorption, emission of particles

diffusion arising from very short range nuclear forces

neutron wavelength much longer than nucleus size

can’t solve neutron structure

Neutrons interact with electrons:

magnetic interactions dipole-dipole

Neutrons have no charge, but they interact (extremely weakly) with charges/electrical fields spin-orbit/Schwinger scattering

see non-resonant magnetic X-ray scattering

‘interactions’ lead to ‘scattering’

Strength of interactions? Measured through ‘scattering cross sections’

In the case of neutrons, nuclear and magnetic scattering are ‘equivalent’

NEUTRON SCATTERING: WHAT DO WE MEASURE?


Scattering is measured in terms of cross-sections

neutron counter set up (θ,Φ) with area dS

at a distance r from ‘scattering system’ large compared compared to sample ‘dimensions’

applies to all types of scattering events

d 2σdΩ dE

=number of neutrons scattered per second into dΩ & dE

Φ dΩ dE

dσdΩ

=number of neutrons scattered per second into dΩ

Φ dΩ

σ = number of neutrons scattered per second / Φ an area (barns)

Φ = number of incident neutrons per area per second

we have ignored the initial and final neutron spin states – to be seen later …..

HOW NEUTRONS INTERACT WITH MATTER – NUCLEAR SCATTERING


nuclear scattering from a single (fixed) nucleus

range of nuclear forces << neutron wavelength

point-like scattering s-wave scattering

elastic scattering (fixed nuclei and no change in nucleus)

same velocity before and after scattering

no absorption (far away from resonance)

spherical scattered wave b : scattering length

of the order of nucleus’s ‘size’

number of neutrons per second into dΩ

cross sections b expressed in fm 10-15m scattering amplitude does not vary with scattering angle

incident flux = neutron density x velocity = v = neutron density = incident plane wave

hmn

k

ei k i .x2

HOW NEUTRONS INTERACT WITH MATTER – NUCLEAR SCATTERING


nuclear scattering from an assembly of nuclei: atom at incident wave: scattered wave at :

ei k i .R i

eik i ⋅Ri −biei k f . r - Ri( )

r -Ri

Ri

Ri

Ψscatt = eik i ⋅Ri −biei k f . r - Ri( )

r -Ri

i∑

orders of magnitude:

nuclear scattering lengths, b’s, depend on isotope, nuclear eigenstate, and nuclear spin orientation relative to neutron spin

wavevector transfer is defined by

Κ

Κ = ki −kf

dσdΩ

= bi* b ji , j

∑ e-iΚ . R i −R j( )

cross section

dσdΩ

=vdSΨscatt

2

vdΩ=

dSdΩ

ei k f .r b jei k i −k f( ).R j

r -R j

j

∑2

= bi* b ji , j

∑ e-i k i −k f( ). R i −R j( )

beware! X-ray boys use different sign convention! Fourier transform

NUCLEAR SCATTERING – COHERENT/INCOHERENT


coherent and incoherent scattering

consider an assembly of similar atoms/ions – spins/isotopes are uncorrelated at different sites

for a single nucleus where taken as real number

bi = b + δbi

δbi = 0

dσdΩ

= bi* b ji , j averaged over all states

∑ e-iΚ . R i −R j( )

sources of incoherent scattering:

isotopic distribution and nuclear spin

coherent scattering: correlations between different sites

incoherent scattering: correlations on the same site (at different times)

particular to neutron scattering

bib j = b 2 + b δbi +δb j( )+ δbiδb j with unless i=j

δbi2 = bi - b 2

= b2 − b 2

dσdΩ

= b 2 e-iΚ ⋅ Ri −R j( )i, j

∑ + b2 − b 2( )N

σcoh = 4π b 2 σincoh = 4π b2 − b 2( )

δbiδb j = 0



If single isotope and zero nuclear spin, no incoherent scattering

b =1

2I+1I+1( )b+ +Ib−[ ]

If single isotope and non-zero nuclear spin I

nucleus+neutron spin: I+1/2 and I-1/2 scattering length b+ and b-

If neutrons and nuclei are un-polarised:

probability ‘plus’ probability ‘minus’

f + =I+12I+1

f + =I

2I+1

b2 − b 2 =I I+1( )2I+1( )2 b+ −b−( )2

To reduce incoherent scattering (background):

use isotope substitution

use zero nuclear spin isotopes

polarise nuclei and neutrons



examples of scattering lengths:

neighbouring elements can be easily identified

NUCLEAR SCATTERING – SCATTERING LENGTHS


most neutron scattering lengths are positive

(same for X-rays)

phase changes by after scattering

no change in phase at scattering point

positive b

negative b e.g. hydrogen

NUCLEAR SCATTERING – CONTRAST VARIATION


useful for imaging and scattering (small angle neutron scattering, liquids)

Mixing 1H and 2H (H and D) allows contrast variation

• full contrast : external shapes

• intermediate contrast : details

100 % H2O 100 % D2O

SCATTERING LENGTHS


Orders of magnitude - numbers for neutron scattering

cross sections are in ~ barns 1 barn = 10-24 cm2

area per atom ~ 10 Å2 = 10 108 barns

1 atom gives 10-9 probability scattering when beam hits it!

to obtain 1% scattering (over 4π) requires 107 layers of atoms

~ 0.1 cm of sample!

Take a single atom of C: b = 6.65 pm = 6.65 10-15m = 6.6510-13cm

σ = 5.56 barns

The probability to observe a single scattered neutron per second requires a huge flux:

I0 =Φ σ Φ =I0 /σ =1/ 5.5610−24( )≈1.81023 particules /cm2 /sec

actual neutron flux ≈ 107 n/cm2/sec, we must either wait for

1.81016 sec ≈ 570 million years or use ~2 1016 atoms ~ 0.4 µg only

SCATTERING LENGTHS


Numbers for neutron scattering

typical neutron flux ~107 n/cm2/sec

sample volumes in the fraction of cm3 range

counting time for ‘incoherent scattering’ from Vanadium (σ ~ 5 barns)

sample volume 1x1x0.1 cm3 i.e. ~ 8.7 1021 atoms

count rate ~ 4 105 n/sec over 4π

detector angular aperture ~ 1% leads to ~ 4 103 n/sec Questions about statistics: • experimental data are ‘counts in the detector’, independent events but with a fixed probability (scattering cross sections!): Poisson’s like

• usual goal is to achieve 1% error per information unit:

• requires ~10,000 counts per bin

• i.e. ~ 0.5 -10 minutes for typical elastic peak ( )

• i.e. at least 10 times longer for inelastic studies ( )

dσdΩ

d 2σdΩ dE

SCATTERING LENGTHS - CONSEQUENCES


Absorption

essentially neutron capture

random variation with

atomic number and isotopes

Applications for detection

neutrons are captured by nuclei

capture creates charged particles

recoiling particles ionise gaseous materials

23He + 0

1n → 13H +p + 0.764 MeV

510B + 0

1n → 37Li + 2

4He + 2.3 MeV

KEY MESSAGES


• neutrons interact with nuclei

• random variation of b’s with atomic number

• isotropic scattering amplitude

• contrast and isotopic substitution

• low absorption

• coherent and incoherent scattering



Imaging with neutrons

selectivity of neutrons – selective imaging through absorption

direct transmission through macroscopic

the branch of an Arauc tree

a piece of rock from the Antartic



Refractive index for neutrons

For a single nucleus, Fermi pseudo-potential

Inside matter, scattering length density

V r( ) =2πh2

mr

b δ r( )

V =2πh2

mρ

ρ =1

volumebi

i∑

∇2 +2mh2 E − V ( )

Ψ r( ) = 0neutrons obey Schrödinger’s equation

In vacuo,

V = 0 and E = Ecin ki2 = 2mE /h2

In the medium,

kf2 = 2m E- V ( )/h2 = ki

2 − 4πρ

n = kf /ki ≈1− λ2 ρ /2π

With b>0, n<1 and neutrons are externally reflected by most materials



Applications

neutron guides: critical angle Ni (Ni58)

γc ≈ λ ρ /π

γc ≈ 0.1Þ−1

N =10−24 2.66 /60.08( )NAvogadro = 0.0267−3

This works with very cold neutrons: λ > 864 Å v~ 4.6 m/s E~ 0.1µeV ~ 1mK!!!

ρ = N bSi + 2bO( )= 4.2110−6 −2

ki2 = 4πρ or λ = π

ρNeutron bottles: a bottle imposes n = o, example SiO2 density: 2.66 molecular weight: 60.08

interference is destroyed if ∆>λ/2 ∆

xc

∆max = λ/2π =1/ki

HOW TO ‘ACCUMULATE’ INTENSITIES?


So far, we have added individual scattering intensities. How to combine them? neutron sources are chaotic: emission over 4π, at ill-defined times with wide distribution of energies, neutrons are moderated

interference pattern from sample to detector

xc

typical values: xc ~ yc ~10 nm (X-ray tomography with pin-hole source xc~500µm)

zc = λ2 /∆λ ≈ 50 × λ ≈10 nm

L

xc

∆

xs

xc /∆max = L /xs xc = λ/2π( )L /xs = 1/ki( )L /xs

source

Do we have to take the neutron source into account?



In general, ignore coherence of neutron sources

and focus on constructive interference of scattered waves

particular case: small angle neutron scattering

what is the largest object that can be measured?

ILL D11

size given by the lateral coherence length

L~40 m, λ = 12Å, xs~10-2 m (adjustable) xc ~ 1µm

xc = λ/2π( )L /xs

Typically, objects smaller than 1µm are studied by scattering methods

objects larger than 1µm are ‘imaged’



Assume plane waves for incident neutrons

‘far away’ from source

source

interference pattern in front of detector

plane waves in scattering system

interference pattern in front of detector

spherical waves emitted by scattering centres

How to combine scattered waves (amplitudes and phases)?



Different ways to combine scattered waves

Born approximation – kinematic theory

neutron wavefunction un-perturbed inside sample

in general OK, away from Bragg reflections and total reflection

Dynamical theory of scattering

takes into account of the change in the neutron wave in the system

see refractive index – but generalised for all scattering vectors

important near total reflection

may be needed near Bragg reflections from perfect crystals with highly collimated beams

In most cases, kinematic theory applies for neutrons



Born approximation, i.e. first order perturbation method

The incident wave travel unperturbed in scattering system

scattering centres with potential V(r’)

dσdΩ

=

Wk i ,λ i →k f ,λ fk f in dΩ∑

Φ dΩFrom the definition,

neutrons scattering system

Wk i ,λ i →k f ,λ fk f in dΩ∑Use Fermi’s Golden rule to calculate transition probabilities

Wk i ,λ i →k f ,λ fk f in dΩ∑ =

2πh

ρk fkf λf V ki λi

2

where is the density of k-states in dΩ per unit energy range

ρk f



some algebra and manipulations as incident wavelength - plane wave in normalisation box

neutron states are periodic in k-space, unit cell volume

vk = 2π( )3

ρk fdEf =

1vk

kf2 dkf dΩ is the number of n-states in dΩ between Ef and Ef+dEf

dEf =h2

mn

kf dkfkinetic energy: therefore:

ρk f=

12π( )3 kf

mn

h2 dΩ

neutron flux

Φ =h

mn

k

scattering cross section with ki, λi and λf fixed

dσdΩ

λ i → λ f

=kf

ki

mn

2πh2

2

kf λf V ki λi

2

energy of neutrons+ scattering system must be conserved

Ei +Eλ i=Ef +Eλ fi

d2σdEf dΩ

λ i → λ f

=kf

ki

mn

2πh2

2

kf λf V ki λi

2δ Ei −Ef +Eλ i

−Eλ f( )partial differential cross section for all scattering potentials

DIFFERENTIAL SCATTERING CROSS SECTION – NUCLEAR SCATTERING


more algebra and manipulations

insert a potential V – Fermi pseudo-potential - short range, scalar and central

V has the form

V = Vjj

∑ r − R j( )= Vjj

∑ x j( ) with x j = r − R j

kf λf V ki λi = χλ f

*∫j

∑ exp −ikf ⋅ r( )V x j( )χλ iexp iki ⋅ r( )dR1dR2dR3 ....dR j...dr

kf λf V ki λi = Vjj

∑ Κ( ) λf exp iΚ ⋅ R j( )λi where

Vj Κ( ) = Vj x j( )exp iΚ ⋅ x j( )∫ dx j

Κ = ki −kf

λf exp iΚ ⋅ R j( )λi = χf*∫ exp iΚ ⋅ R j( )χf

* dR1dR2dR3...dR j ..

Fermi pseudo-potential

Vj x j( )=2πh2

mb j δ x j( ) Vj Κ( ) =

2πh2

mb j

d2σdEf dΩ

λ i → λ f

=kf

ki

b j λf exp iΚ ⋅ R j( )λij

∑2

δ Ei −Ef +Eλ i−Eλ f( )



even more algebra and manipulations

introduce time and energy to reach thermodynamics

d2σdEf dΩ

λ i → λ f

=kf

ki

b jb j ' λi exp -iΚ ⋅ R j '( )λf λf exp iΚ ⋅ R j( )λijj '∑ ×

12πh

exp i Eλ f−Eλ i( )t /h exp -iωt( )dt

-∞

+∞

∫

δ Ei −Ef +Eλ i−Eλ f( )=

12πh


-∞

+∞

∫ hω =Ei −Ef

d2σdEf dΩ

λ i → λ f

=kf

ki

12πh

b jb j 'jj '∑

× λi exp -iΚ ⋅ R j '( )λf λf exp iHt /h( )exp iΚ ⋅ R j( )exp -iHt /h( )λi exp -iωt( )dt-∞

+∞

∫where H is the Hamiltonian of the scattering system

Introduce time-dependent operators

R j t( ) = exp iHt /h( )exp iΚ ⋅ R j( )exp -iHt /h( )

Introduce partition function , probability



To get measured cross section, sum over all final λf (with fixed λi) and average over all λi.

Z = exp −Eλ /kBT( )λ

∑

δ Ei −Ef +Eλ i−Eλ f( )=

12πh


-∞

+∞

∫ hω =Ei −Ef

pλ =1Z

exp −Eλ /kBT( )

d2σdEf dΩ

= pλ i

d2σdEf dΩ

λ i → λ fλ iλ f

∑

=kf

ki

12πh

b jb j 'jj '∑ exp -iΚ ⋅ R j ' (0) exp iΚ ⋅ R j (t) exp -iωt( )dt

-∞

+∞

∫

A compact form, but not easy to calculate:

measure of ‘pair correlation functions’

information on scattering system contained in time-dependent ops.

and wavefunctions

no correlation between b’s on different sites



Consider a simple system with a single element but different b’s

b jÕb j = b 2 , j '≠ j b jÕb j = b2 , j '= j

d2σdΩ dEf

coh

=σcoh

4πkf

ki

12πh

exp -iΚ ⋅ R j ' (0) exp iΚ ⋅ R j (t) exp -iωt( )dt-∞

+∞

∫jj '∑

Coherent scattering: correlation between the position of the same nucleus at different times and correlation between the positions of different nuclei at different times

interference effects

d2σdΩ dEf

=kf

ki

12πh

b jÕb j j ', j exp -iωt( )dt-∞

+∞

∫jj '∑ b = fi bi

i∑ b2 = fi bi

2

i∑

d2σdΩ dEf

incoh

=σincoh

4πkf

ki

12πh

exp -iΚ ⋅ R j (0) exp iΚ ⋅ R j (t) exp -iωt( )dt-∞

+∞

∫j

∑

Incoherent scattering: only correlation between the position of the same nucleus at different times

no interference effects

KEY MESSAGE


• neutron scattered intensities are proportional to space and time Fourier transforms of site correlation functions

KEY MESSAGE

9 July 2014 l Christian Vettier ESRF-Grenoble & ESS-Lund 20 Octobre 2010 40 ans après le prix Nobel de Louis Néel C. Vettier 42

BES DOE http://www.nano.gov/html/facts/The_scale_of_things.html

neutrons cover a wide range of length scales

imaging/scattering

CRYSTALLINE MATERIALS


Examples of objects on a lattice - crystalline/ordered materials

Real space

1-d system:

convolution of objects and lattice of Dirac functions

N objects, N large

Reciprocal space

1-d system:

Fourier transforms

For N large

Similarly we define associated reciprocal spaces that reflect the symmetry and periodicities of real space lattices

NUCLEAR SCATTERING FROM CRYSTALLINE MATERIALS


Crystalline materials:

all atoms (nuclei) have an equilibrium position and they move about it

atom in cell j: can be generalised to non-Bravais lattices

R j t( ) = j + u j t( )

exp -iΚ ⋅ R j ' (0) exp iΚ ⋅ R j (t) jj '∑ = N exp iΚ ⋅ j( )

j∑ exp -iΚ ⋅ u0(0) exp iΚ ⋅ u j (t)

exp -iΚ ⋅ R j (0) exp iΚ ⋅ R j (t) j

∑ = N exp -iΚ ⋅ u0(0) exp iΚ ⋅ u0(t) displacements u(t) can be expressed in terms of normal modes or phonons

q

q

q



exp -iΚ ⋅ R j ' (0) exp iΚ ⋅ R j (t) jj '∑ = N exp iΚ ⋅ j( )

j∑ exp -iΚ ⋅ u0(0) exp iΚ ⋅ u j (t)

Coherent part exp V exp U

it can be shown (Squires)

expU expV = exp U2 exp UV

d2σdΩ dEf

coh

=σcoh

4πkf

ki

N2πh

exp U2 exp iΚ ⋅ j( ) exp UV exp -iωt( )dt-∞

+∞

∫j

∑

Debye-Waller factor

2W = − U2 = K⋅ u 2

d2σdΩ dEf

coh

=σcoh

4πkf

ki

N2πh

exp −2W( ) exp iΚ ⋅ j( ) 1+ UV +12!

UV 2 +.....

exp -iωt( )dt

-∞

+∞

∫j

∑

zero-th order: coherent elastic scattering - Bragg scattering

1st order: coherent one-phonon scattering

……



Coherent elastic scattering diffraction

purely elastic scattering

d2σdΩ dEf

coh

=σcoh

4πkf

ki

N2πh

exp -2W( ) exp iΚ ⋅ j( ) exp -iωt( )dt-∞

+∞

∫j

∑

N: number of unit cells

v0: volume of the unit cell

exp -iωt( )dt-∞

+∞

∫ = 2πh δ hω( )

ki = kf

d2σdΩ dEf

coh el

=σcoh

4πN exp -2W( ) exp iΚ ⋅ j( )δ hω( )

j∑

dσdΩ

coh el

=d2σ

dΩ dEf

coh el

dEf =0

∞

∫ σcoh

4πN exp -2W( ) exp iΚ ⋅ j( )

j∑

dσdΩ

coh el

=σcoh

4πN

2π( )3

v0

exp -2W( ) δ Κ −τ( )τ

∑

dσdΩ

coh el

= N2π( )3

v0

δ Κ −τ( )τ

∑ FN Κ( ) 2FN Κ( ) = bd

d∑ exp iΚ ⋅ d( )exp −Wd( )

non-Bravais lattice with different atoms d in the unit cell

structure factor



Bragg’s law

λ = 2d sinθ

Collecting intensities at Bragg peaks gives access to ‘squared’ values of Fourier components of the structure

Collect as many as possible to overcome the phase problem

Practical application

monochromators!



Coherent elastic scattering (diffraction) provides:

• periodicity in space, lattice symmetry and lattice constants

• positions of atoms in cells from

• powder and single crystals methods

But requires inversion of intensities into phases/amplitudes

Self propagating synthesis

High temperatures

Phase transitions visible

FN Κ( )2

CRYSTALLINE MATERIALS : DIFFRACTION INSTRUMENTS


How does it work?

neutron source

monochromatisation

crystal – diffraction

or time of flight

detector

Detector 2D

Detector 1D

CRYSTALLINE MATERIALS : BEAM FILTERS


Maximum wavelength for which no Bragg scattering can occur

λmax = 2dmax sinθ( )max = 2dmax

Bragg cut-off is the maximum plane spacing

dmax

Beyond λmax σtot drops

σtot = σscatt + σabsor

Cut-off wavelengths for polycrystalline materials

Be: 3.9 Å graphite: 6.7 Å

Transmission of pyrolytic graphite

KEY MESSAGES


• neutron diffraction is an essential tool for structures determination

• it complements other diffraction methods

• neutrons probe bulk samples



Inelastic coherent scattering (Bravais lattice)

d2σdΩ dEf

coh

=σcoh

4πkf

ki

N2πh

exp −2W( ) exp iΚ ⋅ j( ) 1+ UV +12!

UV 2 +.....

exp -iωt( )dt

-∞

+∞

∫j

∑

one-phonon coherent scattering (Bravais lattice) <UV> involves creation and annihilation of phonon modes

UV =h

2MNΚ ⋅ es( )

ωss∑ exp -i q⋅ j− ωst( ) ns +1 +exp i q⋅ j− ωst( ) ns[ ]

d2σdΩ dEf

coh ±1

=σcoh

4πkf

ki

2π( )3

v0

12M

exp −2W( )s∑ Κ ⋅ es( )

ωs

ns +1/2 ±1/2τ

∑

× δ ω mωs( )δ Κ mq − τ( )

ki

kf

Κ

ki < kf

energy gain

ki

kf

Κ

ki > kf

energy loss

Κ = τ ± q

Ei −Ef = hω = ± hωs

conservation laws!

NUCLEAR INELASTIC SCATTERING


q

q

q

How to measure?

Orient sample: sample frame in coincidence with instrument frame

Go to a given

Κ = τ ±q and energy transfer hω

Collective excitations – phonons (magnons)

PHONON MODES


classical instruments: triple-axis machines

applications:

lattice interactions, ‘soft mode’

phase transitions

superconductivity, …

scan point by point

either constant ‘K’

or constant ‘energy’

constant ‘K’

SPECTROSCOPY


Spectroscopy – internal modes – little dispersion of modes

Excitations in hypothetical molecular crystal J.Eckert SpectroChim. Acta 48A, 271 (1992)

Use incoherent scattering

d2σdΩ dEf

incoh ±1

=kf

ki

δ ω mωs( ) ns +1/2 ±1/22ωss

∑σincoh( )r

4π1

Mr

K⋅ er2 exp −2Wr( )

r∑

SPECTROSCOPY


more global picture: time of flight and large detector coverage

TIME DOMAIN


Time domain is reached through inelastic scattering

the range of energy transfer that can be covered determines the ‘accessible’ time domain (Fourier transform!)

Depends on ‘incident’ neutron energy

and instrumental resolution

Typical resolution in wavelength (and energy) are ~ a few percents

except special techniques (backscattering, NSE)

TIME DOMAIN AND COMPLEMENTARITY


complementarity with other methods

KEY MESSAGES


• accessible time and space domains cover a wide range of applications

• no single probe can cover the whole (K,ω) space that would allow an ideal Fourier transformation

• use complementary probes

DISORDERED MATERIALS


liquids, glasses, ..no equilibrium positions for atoms

local order but no long range order

exp -iΚ ⋅ Ri − R j( ) =sin Krij( )

Krij

consider mono-atomic system –

dσdΩ

= bi* b ji , j

∑ e-iΚ . R i −R j( )

for most liquids, glasses, ... scattering depends on magnitude of

averaging over polar angles

replace sum over atoms by radial distribution function

g(r) for monoatomic liquid

DISORDERED MATERIALS - LIQUIDS


note that

dσdΩ

= N b2 + bi b j e-iΚ . R i −R j( )

i ≠ j

N

∑ = N b2 +4πρ b2 r 2 g r( )sin Kr( )Kr( )0

∞

∫ dr

mono-atomic liquid

r 2 sin Kr( )Kr( )0

∞

∫ dr = 0 unless K = 0

dσ K ≠ 0( )dΩ

= N b2 S K( ) with = S K( )=1+4πρK

r g r( )−1[ ] sin Kr( )0

∞

∫ dr

S(K): structure factor

very different from FN(K) in crystalline materials

• relates to intensity (not amplitude)

• not well structured, only a few peaks

Liquid Argon

T= 85K

Experimental data

Inverted ρ(r)

(Fourier transform) J.L. Yarnell et al.

PRA 7, 2130 (1973)

DISORDERED MATERIALS - LIQUIDS


In n-component systems, there are n(n+1)/2 site-site radial distributions

To be measured, using isotopic substitution.

In liquids, strictly speaking there is no elastic neutron scattering:

nuclei recoil under neutron impact

need to span all (K,ω) space

or apply inelasticity corrections Placzek PRB 86, 377 (1956) Typical instruments

ILL D4

Sandals ISIS

MAGNETISM!


Neutrons have a spin ½ and therefore carry a magnetic moment

neutron beams can be polarised

µn = −γ µNσ where µN =eh

2mp

γ =1.913

µe = −2µB s where µB =eh

2me

For comparison, electrons have

Neutron magnetic moments feel magnetic fields created in materials:

- electrons: dipole moments and currents

- nuclei: dipole moments (neglected here)

d2σdEf dΩ

λ i → λ f

=kf

ki

mn

2πh2

2

kf λf V ki λi

2δ Ei −Ef +Eλ i

−Eλ f( )The potential V in the scattering cross section should include these effects

MAGNETISM!


Β = ΒS + ΒL =µ0

4πcurl

µe × RR2

−

2µB

hp × RR2

Magnetic field created at distance R electron with momentum p

Potential of a neutron in B

Vm = − µn⋅ B =−µ0

4πγ µN 2 µB curl

s × RR2

+

1h

p × RR2

d2σdEf dΩ

σ iλi →σ f λf

=kf

ki

mn

2πh2

2

kfσf λf Vm kiσi λi

2δ Ei −Ef +Eλi

−Eλf( )

During scattering, neutron changes from state ki,σi to kf,σf

Complex evaluation:

magnetic interaction is long range

magnetic forces are not central

MAGNETISM!


d2σdEf dΩ

σ iλ i →σ f λ f

=kf

kiγ r0( )2

σf λf σ⋅ Q⊥ σi λi

2δ Ei −Ef +Eλ i

−Eλ f( )

whereas for nuclear scattering

Q⊥ = exp iΚ ⋅ rx( )electrons x

∑ ˆ Κ × sx × ˆ Κ ( )+ ihK

px × ˆ Κ ( )

b j exp iΚ ⋅ R j( )j

∑

After long calculations

M(K) is the Fourier transform of M(r)

Q⊥ Κ( ) = ˆ K × Q Κ( )× ˆ K ( ) where Q Κ( )=−1

2µB

M Κ( ) ˆ K is a unit vector

The operator is related to the magnetisation of the scattering system

separating spin and orbital contributions

Q⊥

is the vector projection of Q on to the plane perpendicular to K

Q⊥

‘strength’ of scattering ~ (γr0)2 of the order of 0.3 barn r0: classical radius of electron 2.810-13 cm

KEY MESSAGES


• interactions with electrons connect to magnetisation densities

• ‘magnetic’ scattered intensities are proportional to space and time Fourier transforms of site correlation functions for magnetic moments

• ‘nuclear’ and ‘magnetic’ interactions have similar strengths

MAGNETISM!


Similarly to nuclear scattering, magnetic neutron scattering probes ‘correlations’

d2σdEf dΩ

=

kf

ki

γr0( )2

2πhδαβ − ˆ K α ˆ K β( ) Qα −Κ ,0( )Qβ Κ ,t( )∫

αβ

∑ exp −iωt( )dt

Qβ Κ ,t( ) = exp iHt /h( )Qβ Κ( )exp −iHt /h( )

equivalent to nuclear scattering

geometrical factor

Elastic scattering – thermal average at infinite time

dσdΩ

el

= γr0( )2δαβ − ˆ K α ˆ K β( ) Qα −Κ( )Qβ Κ( )

αβ

∑

or

dσdΩ

el

=γr02µB

2

ˆ Κ × M Κ( ) × ˆ Κ 2

with

Q Κ( )=−1

2µB

M Κ( )

contains all information on magnetic arrangements

symmetry, periodicity, moments, …..

M Κ( )

MAGNETIC DIFFRACTION


Ferromagnets

localised magnetic system has the same periodicity as lattice

dσdΩ

el

= γr0( )2N

2π( )3

v0

Sη 2 12

gF τ( )

τ

∑2

exp −2W( ) × 1− ˆ τ ⋅ ˆ η ( )Aver2 δ Κ − τ( )

- magnetic intensity on top of nuclear intensity

- ‘magnetic form factor’ not constant as b –

spatial distribution of ‘magnetic’ electrons

- Measurements of intensities give F(τ) which allow M(r) to be calculated

Non-ferromagnets

new periodicity in space leads to new Bragg peaks

MAGNETIC DIFFRACTION


Non-ferromagnets

neutrons allow to probe local magnetic order

powder samples or single crystals

‘easy’ and routine experiments!

One of the very strong points for neutrons

C. Shull et al. 1949

More complex materials

Important for new devices

KEY MESSAGE


• neutron diffraction (powder) is the the method of choice to determine magnetic structures (if not the only one …)

INELASTIC MAGNETIC SCATTERING OF NEUTRONS


Inelastic magnetic neutron scattering probes magnetic ‘correlations’

simple localised ‘magnetic’ excitations

(crystal field levels)

spin waves

fluctuations (spin density fluctuations)

interactions between lattice and magnetism

phase diagram:

low T and high H

INELASTIC MAGNETIC SCATTERING OF NEUTRONS


very powerful experimental method

investigating pairing mechanisms in superconductors

phonon-like or magnetic?

Sr3Ru2O7 field-induced QCP

T=40 mK E=0.8 meV

6T

8T

NEUTRON POLARISATION


scattering cross section involves neutron spin states

another neutron degree of freedom

use of polarised neutrons

use of polarisation

use of neutron spin precession in fields

Larmor precession

spin manipulation

spin filters

NSE methods



polarised neutron beams, up (u) and down (v) states

P =n+ − n−

n+ + n−

previous cross-sections gives rise to 4 cross-sections

u →u v →v u →v v →u

coherent nuclear scattering

incoherent nuclear scattering

u →uv →v

b =I +1( )b+ + I b−

2I +1isotopes

u →vv →u

b = 0

u →uv →v

b2 − b 2 =I +1( )b+ + I b−

2I +1

2

isotopes

−I +1( )b+ + I b−

2I +1isotopes

2

+13

b+ − b−

2I +1

2

I I +1( )isotopes

u →vv →u

b2 − b 2 =23

b+ − b−

2I +1

2

I I +1( )isotopes

particular cases: unpolarised neutrons

Ni: all isotopes with I=0

Vanadium: only one isotope



Polarisation ‘induces’ interference between nuclear and magnetic scattering

In ferromagnets, and are non-zero for the same K vectors

dσdΩ

= N2π( )3

v0

FN Κ( )2+ 2 ˆ P ⋅ ˆ µ ( )FN Κ( ) FM Κ( ) + FM Κ( ) 2

FN Κ( )

FM Κ( )

Similar effects/applications in reflectometry – ‘magnetic’ optical index

polarising neutron guides

ˆ P ⋅ ˆ µ ( )= ±1

Κ

guide fields

polarising field application: polarisation devices

dσdΩ

= N2π( )3

v0

FN Κ( ) ± FM Κ( )2

if matching FN and FM reflected beam is polarised

for neutrons (anti-)parallel to B

B ↑



dσdΩ

= N2π( )3

v0

FN Κ( ) ± FM Κ( )2

application: precise measurement of weak magnetic signals

apply B perpendicular to K

moments are aligned parallel to B

measure flipping ratio R

R =dσdΩ

+

/dσdΩ

−

=1− γ1+ γ

2

with γ = FM Κ( )/FN Κ( )

if γ is small, R~1-4γ

allows to measure spin densities

Iron C.G Shull et al. J.Phys.Soc.Japan 17,1 (1962)

SO MANY OTHER FIELDS


neutron spin-echo: use of Larmor precession of neutron’s spin

neutron reflectometry, SANS, …

time evolution of s=1/2 in magnetic field B

total precession angle depends neutron’s velocity

with B=10 Gauss ~29 turns/m for 4Å neutrons

dr s

dt= γ

r s ×

r B ωL = γ B with γ= - 2913∗2π Gauss-1⋅ s−1

φ = ωL t = γ B d /v

neutron spin-echo encodes neutron velocity – quite high resolution

without loss in intensity

NSE breaks the awkward relationship between intensity and resolution: the better the resolution, the smaller the resolution volume and the lower the count rate!

SUMMARY OF KEY MESSAGES


• neutrons have no charge – low absorption

• ‘nuclear’ and ‘magnetic’ interactions have similar strengths

• interaction with nuclei very short range

isotropy, isotope variation and contrast

• interactions with electrons lead to magnetisation densities

neutron diffraction the method of choice to determine magnetic structures

• scattered intensities are proportional to space and time Fourier transforms of site correlation functions (positions and magnetic moments)

• accessible time and space domains cover a wide range of applications

• caveat: neutron sources are not very efficient …..

FURTHER READING


Introduction to the Theory of Thermal Neutron Scattering

G.L. Squires Reprint edition (1997) Dover publications ISBN 04869447

Experimental Neutron Scattering

B.T.M. Willis & C.J. Carlile (2009) Oxford University Press ISBN 978-0-19-851970-6

Neutron Applications in Earth, Energy and Environmental Sciences

L. Liang, R. Rinaldi & H. Schober Eds Springer (2009) ISBN 978-0-387-09416-8

Methods in Molecular Biophysiscs

I.N. Serdyuk, N. R. Zaccai & J. Zaccai Cambridge University Press (2007) ISBN 978-0-521-81524-6

Thermal Neutron Scattering

P.A. Egelstaff ed. Academic Press (1965)

european synchrotron radiation facility · why neutrons? page 2 9 july 2014 l christian vettier...

Documents